L(s) = 1 | + 22i·2-s + 81i·3-s + 28·4-s − 1.78e3·6-s − 5.98e3i·7-s + 1.18e4i·8-s − 6.56e3·9-s − 1.46e4·11-s + 2.26e3i·12-s − 3.79e4i·13-s + 1.31e5·14-s − 2.47e5·16-s − 4.41e5i·17-s − 1.44e5i·18-s − 4.41e5·19-s + ⋯ |
L(s) = 1 | + 0.972i·2-s + 0.577i·3-s + 0.0546·4-s − 0.561·6-s − 0.942i·7-s + 1.02i·8-s − 0.333·9-s − 0.301·11-s + 0.0315i·12-s − 0.368i·13-s + 0.916·14-s − 0.942·16-s − 1.28i·17-s − 0.324i·18-s − 0.777·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.21502 - 0.286827i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21502 - 0.286827i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 81iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 22iT - 512T^{2} \) |
| 7 | \( 1 + 5.98e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 1.46e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.79e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 4.41e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + 4.41e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.26e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 1.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.91e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.09e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 1.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.31e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 1.38e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 - 5.76e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 3.20e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.10e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.18e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 2.76e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.64e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 4.48e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.51e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 1.89e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.01e9iT - 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.74594626813386424008245941076, −11.18118239488921377651088810580, −10.44890394265082589090948200058, −8.986513306100013007822394697892, −7.75327791669520941189934943986, −6.80336810253944511128318883854, −5.48028155969253773839515635547, −4.27320938169622283223041379902, −2.54397704664273585491582810438, −0.33832489089603146709014820786,
1.46165159683835399454857114661, 2.34731343852841413103705855525, 3.70580092723797461145831204968, 5.64962045413972067982749364013, 6.85437863371775176420743866317, 8.300180848530345090116581746499, 9.535362288367991398193201668266, 10.80053147684857431110336739471, 11.71701503042245599847971439042, 12.56684008616730797580044102699